Method, computer program product, and system for risk management

ABSTRACT

Described are a method, computer program product, and system for risk management using readily available, gridded hazard data to estimate and obtain a risk analysis parameter (e.g., expected repair cost) for use in risk management, such as in seismic risk management. The method includes calculating economic risk for buildings in terms of an expected annualized loss (EAL). EAL is the product of a scenario loss estimate called probable frequent loss (PFL) and an economic hazard coefficient (H). H can be created using readily available gridded hazard data produced by the U.S. Geological Survey. The method also includes a technique for calculating shaking intensity, s EBE , which is needed for determining PFL. Incorporated into a system, the system can be utilized by engineering consultants (or others interested in risk management) via the Internet, or on any other computer readable medium.

PRIORITY CLAIM

This application is a Continuation In Part Application, claiming the benefit of priority to non-provisional application number 10/862,185, filed in the United States on Jun. 4, 2004, and titled “Method, Computer Program Product, and System for Risk Management,” and also claiming the benefit of priority to provisional application number 60/585,260, filed in the United States on Jul. 01, 2004, titled, “A Method and Software Application for Calculating the Site Economic Hazard Coefficient and Economic-Basis Event Shaking Intensity from Gridded Hazard Data.”

BACKGROUND OF THE INVENTION

(1) Technical Field

The present invention relates to risk management. More specifically, the present invention relates to a method, computer program product, and system for employing readily available, hazard data to estimate an expected repair cost for use in risk management, such as in seismic risk management.

(2) Description of Related Art

The field of seismic risk management has been gradually developing over the past few decades, increasingly enabled by technological advances in software and driven by a need for more informed property ownership decisions.

Seismic risk enters into several important real estate decision-making processes, such as the purchase of investment property, performance-based design of new structures, seismic rehabilitation of existing buildings, and decisions regarding the purchase of earthquake insurance. In such situations, example of important factors include who the decision-makers are, how they make decisions, what aspects of seismic risk most concern them, and the length of their planning horizon.

Economic seismic risk to large commercial properties in seismically active regions with commercial mortgages is assessed every time the property changes hands, typically on the order of every five to ten years. By contrast, a building is designed and built only once. Thus, the most common opportunities for market forces to bring about seismic-risk mitigation for commercial properties are at times of sale. Anecdotal evidence suggests that these are mostly missed opportunities, as risk is typically not mitigated, even in more vulnerable buildings.

This can be partly explained by considering the context in which seismic assessments are performed. During virtually every sale of an existing commercial building, the buyer assesses the building's investment value using a financial analysis that considers revenues and expenses, rent roll, market leasing, physical condition, and other property information. The investor makes his or her bidding decision based on projected income and expenses, using one or more of the economic performance metrics of net present value, net operating income, cash flow, internal rate of return, and capitalization rate.

The input to this financial analysis is typically provided by a real estate broker representing the seller, whose figures the investor checks and modifies during a due-diligence study. Many of the inputs are known values, such as the quantity of leases, duration, and income from current leases. However, many other values are uncertain. Vacancy rates, market rents, and other important parameters fluctuate significantly and unpredictably, leading to substantial uncertainty in the future economic performance of a property. In the face of these uncertainties, the bidder usually estimates investment value using best-estimate inputs and then again with deterministic sensitivity studies to probe conditions that would lead to poor performance (higher future vacancy rates, for example). The future cost to repair earthquake damage is not one of the parameters the bidder uses in the financial analysis. This is important because seismic risk is not a market quantity.

The real estate market is not wholly without forces to influence seismic-risk mitigation. The due-diligence study typically includes an engineering assessment of the condition of the property, which itself typically includes an estimate of the earthquake probable maximum loss (PML). PML is by far the dominant earthquake risk parameter in financial circles.

The earthquake PML has no standard quantitative definition. Most working definitions involve the level of loss associated with a large, rare event. One definition is that PML is the 90^(th) percentile of loss given the occurrence of what building codes until recently called the design basis earthquake (DBE). The DBE is an event producing a shaking intensity with 10% exceedance probability in 50 years. Colloquially (and inexactly), this is an upper-bound loss given the 500-year earthquake. More accurately, assuming Poisson arrivals of earthquakes, this shaking level has a mean occurrence rate of 0.00211 yr⁻¹ and a mean recurrence time of 475 years. Because this PML is the 90^(th) percentile loss given this level of shaking, the PML-level loss can have a much longer mean recurrence time.

Commercial lenders often use PML to help decide whether to underwrite a mortgage. It is common, for example, for a commercial lender to refuse to underwrite a mortgage if the PML exceeds 20% to 30% of the replacement cost of the building, unless the buyer purchases earthquake insurance, a costly requirement that often causes the investor to decide against bidding. Once the PML hurdle is passed, the bidder usually proceeds to ignore seismic risk, for at least the following:

1. Irrelevant planning period. Investors plan on the order of 5 years, making loss corresponding to shaking intensity with a 500-year recurrence time largely irrelevant, too rare even for consideration in a sensitivity study.

2. Incompatibility with financial analysis. PML is a scenario value, not an ongoing cost that can be reflected in a cashflow analysis.

3. Custom. Investors are not required by custom or regulation to include seismic risk in the financial analysis.

Lacking any measure of economic risk beyond PML, the bidder has no basis for assessing how the seismic risk of a building should influence the purchase price or for judging whether seismic risk mitigation might be worth exploring. Faced with a high PML, the bidder might increase the discount rate used in the financial analysis to reduce the present value of the future net income stream.

Improving upon the prior art in order to increase the efficacy of seismic risk analysis, the Applicant previously filed application number 10/862,185, entitled, “Method, Computer Program Product, and System for Risk Management (hereinafter Application '185).” The prior application discloses a method for the calculation of economic seismic risk for buildings in terms of expected annualized loss (EAL) and a scenario loss estimate called probable frequent loss (PFL), both of which measure a building owner's risk of earthquake-damage repair costs.

Application '185 disclosed that PFL can be reasonably estimated using a simplified performance-based earthquake engineering (PBEE) analysis that involves a single linear structural analysis and some simple calculations of loss conditioned on structural response.

It was shown that: EAL≈H·PFL,   (1) where $\begin{matrix} {{H \equiv \frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}},} & (2) \end{matrix}$ in which EAL is defined as the expected annualized value of repair cost to a particular building in a particular location and H is referred to as the economic hazard coefficient. The PFL is a scenario loss estimate: the estimated mean building repair cost conditioned on the occurrence of shaking in the economic-basis event, EBE. The EBE is defined as the event producing site shaking with 10% probability of exceedance in 5 years (Compare with the design-basis earthquake, DBE, whose shaking intensity has 10% exceedance probability in 50 yr.). The site shaking intensity of the EBE is denoted by s_(EBE). G_(EBE) is the mean exceedance frequency at that site of the shaking intensity s_(EBE). G_(NZ) refers to the mean exceedance frequency of the lowest shaking intensity s_(NZ) that would produce a nonnegligible repair cost. The intensities s_(NZ) and s_(EBE) are measured in terms of damped elastic spectral acceleration at period T and damping ratio ζ.

Note that the methodology also produces reasonable estimates of EAL if EBE is defined slightly differently. For example, it was found that similar results are produced for EBE defined as the event producing site shaking with a 50% probability of exceedance in 50 years. The value of defining EBE as is done here (10% in 5 yr, rather than 50% in 50 yr) is for its relevance to the typical commercial real-estate investor, for whom 50 years is too long a planning period.

The methodology for calculating PFL, H, and EAL is expected to be of value to commercial real-estate investors and possibly other stakeholders of high-value buildings in seismically active regions. For example, it is useful in several identifiable ways:

-   -   a. Relevant scenario loss for the investor 's sensitivity study.         First, the PFL reflects a reasonable upper-bound loss within the         typical investor's planning period (mean loss conditioned on         shaking with 10% exceedance probability in 5 years). This is in         contrast with the commonly-estimated probable maximum loss         (PML), which tends to reflect a loss associated with the DBE.         While shaking and loss with 500-yr or longer return period is         far too rare to be of interest to the typical commercial         investor, PFL could be reasonably employed in the investor's         sensitivity studies. The investor typically examines impact on         investment value (measured, e.g., by return on equity) under         various what-if situations, such as future vacancy rates that         are higher than expected, or market rents that are lower than         expected. The investor would know that PFL could calculate by         how much return on equity is reduced if the EBE occurs.     -   b. Investor can reflect seismic risk as an operating expense.         PFL can be multiplied by H to produce EAL, which can be readily         employed as an operating expense in the investor's financial         analysis. Banks are already beginning to quantify their seismic         risk in terms of EAL; real-estate investors may not be far         behind.     -   c. Short learning curve for investor and engineering consultant.         PFL is conceptually very similar to PML, and should be readily         understood both by investors who already understand PML, and by         engineering consultants who currently estimate PML for         investors. EAL should also be readily understood by investors as         amortized seismic risk. The investor need not understand seismic         vulnerability nor seismic hazard, and the engineering consultant         need not create a complete seismic vulnerability function or         seismic hazard function.     -   d. A new consulting product for little extra work. The engineer         who estimates PML can at the same time estimate PFL using the         same information and procedures. By looking up H on a paper map         or electronic database, and calculating EAL as the product of H         and PFL, the consultant can provide his or her client with         valuable new information for little additional effort. The same         maps and database can, incidentally, provide the consultant with         EBE and DBE shaking intensity.

While the prior invention presented a method to calculate seismic risk management by calculating PFL and EAL, given s_(NZ), G_(NZ), s_(EBE), and G_(EBE), it did not provide a method for calculating G_(NZ), s_(EBE), G_(EBE). Thus, a continuing need exists for a system and methodology for employing readily available, gridded hazard data to calculate G_(NZ), s_(EBE), G_(EBE), and thereby H, for any geographic location and fundamental period (T) within the boundaries of the gridded data.

SUMMARY OF THE INVENTION

The present invention relates to risk management. More specifically, the present invention relates to a method, computer program product, and system for employing readily available, hazard data to estimate and obtain a risk analysis parameter (e.g., estimated repair cost) for use in risk management, such as in seismic risk management.

The method for obtaining a risk analysis parameter using a data processing system having an input and an output, comprises acts of:

-   -   a. querying a user to input at least one of the following input         parameters into the input:         -   i. location ô=({circumflex over (φ)}, {circumflex over             (λ)}), where ô denotes a longitude and latitude of a             facility in question;         -   ii. site classification ĉ, where the site classification             reflects geological characteristics of soil at the facility             in question;         -   iii. period of interest {circumflex over (T)}, where             {circumflex over (T)} denotes a user's estimate of the             small-amplitude fundamental period of vibration of the             facility in question;         -   iv. damping ratio {circumflex over (ζ)}, where the damping             ration is a mathematical characteristic of an idealized             structural system of the facility in question;         -   v. loss-threshold shaking s_(NZ), where the loss-threshold             shaking is a minimum level of shaking at which nonnegligible             damage occurs;         -   vi. planning period t_(EBE), where the planning period is             the period an investor uses in a financial analysis for the             facility in question; and         -   vii. exceedance probability p_(EBE), where the exceedance             probability is the probability that earthquake shaking of             intensity s_(EBE) or greater will occur during planning             period t_(EBE);     -   b. calculating a risk analysis parameter based upon at least one         of the input parameters;     -   c. outputting the risk analysis parameter through the output,         whereby through the risk analysis parameter, a user may         calculate an expected annualized loss (EAL) according to the         following:         EAL=H×PFL,     -   wherein PFL represents a probable frequent loss, average loss         conditioned on a seismic intensity associated with an         economic-basis earthquake, and H represents an economic hazard         coefficient.

In another aspect, in the act of calculating a risk analysis parameter, the risk analysis parameter is hazard coefficient H, and the act of calculating hazard coefficient H further comprises acts of:

-   -   a. calculating G_(EBE), where G_(EBE) is an average exceedance         frequency of s_(EBE), and where s_(EBE) is the seismic intensity         associated with an economic-basis earthquake;     -   b. identifying grid points closest to ô for location         ô=({circumflex over (φ)}, {circumflex over (λ)}), where         {circumflex over (φ)} denotes a latitude designation and         {circumflex over (λ)} denotes a longitude designation, the grid         points closest to {circumflex over (λ)} being referred to as the         bounding grid points o_(j);     -   c. considering the period of interest {circumflex over (T)},         determining a period of at least one hazard curve, referred to         as a bounding period T_(i);     -   d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s))         of s_(r) in the hazard curve G(s_(r)|o_(j)c=c_(r),T=T_(i),ζ=ζ)         and adjusting s_(r,k) for each bounding grid point o_(j) and         bounding period T_(i) to account for site classification ĉ and         for damping ratio {circumflex over (ζ)}, where s is seismic         intensity and where adjusting s_(r,k) is done by multiplying         s_(r,k) by a site-classification coefficient F_(c) and a damping         coefficient F_(ζ), where by denoting by s_(k) the k^(th) value         of s at the same location and period after adjusting for site         classification and damping ratio,         G(s _(k) |o _(j) ,c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c)         F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . .         . n_(s);     -   e. letting G_(ij,NZ) denote the damping- and site-class-adjusted         average exceedance frequency of s_(NZ) given period T_(i) and         location o_(j), and for each T_(i) and o_(j), interpolating each         curve from “Act d” to determine G_(ij,NZ);     -   f. letting G_(j,NZ) denote the damping- and site-class-adjusted         value of G_(NZ) at the period of interest {circumflex over (T)}         and location o_(j), and for each j, calculating G_(j,NZ);     -   g. letting G_(NZ) denote the damping- and site-class-adjusted         value of G_(NZ) at the period of interest {circumflex over (T)}         and location of interest ô, and calculating G_(NZ);     -   h. calculating risk analysis parameter H (i.e., hazard         coefficient) according to the following:         ${H \equiv \frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}},$         whereby through H, a user may calculate an expected annualized         loss (EAL) according to the following:         EAL=H×PFL,         wherein PFL represents a probable frequent loss, mean loss         conditioned on an occurrence of s_(EBE).

In another aspect, in the act of calculating G_(EBE), G_(EBE) is calculated based on t_(EBE) and p_(EBE) assuming Poisson earthquake arrivals according to the following: G _(EBE) =−ln(1−p _(EBE))/t _(EBE),

-   -   where G_(EBE) is a mean annual exceedance frequency of s_(EBE),         and where s_(EBE) is the seismic intensity associated with an         economic-basis earthquake.

In yet another aspect, the act of identifying grid points closest ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), further comprises acts of identifying four grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the four grid points closest to ô being referred to as the bounding grid points o₁=(φ₁, λ₁), o₂=(φ₁+Δφ, λ₁), o₃=(φ₁, λ₁+Δλ) and o₄=(φ₁+Δφ,λ₁,+Δλ)}, where o₁, o₂, o₃, o₄ε{(φ_(r), λ_(r))} such that φ_(min)≦φ₁<{circumflex over (φ)}≦φ₁+Δφ≦φ_(max) and λ_(min)≦λ₁<{circumflex over (λ)}≦λ₁+Δλ≦λ_(max), and calculating x and y according to the following: x=({circumflex over (φ)}−φ₁)/(Δφ) and y=({circumflex over (λ)}−λ₁)/(Δλ).

Additionally, the act of determining a period of at least one hazard curve, further comprises an act of determining periods of two hazard curves, referred to as bounding periods, T₁, T₂ε{T_(r)}, such that T₁<{circumflex over (T)}≦T₂, or T₁={circumflex over (T)} if {circumflex over (T)} is in {T_(r)} and is equal to its minimum value, and T₁ is the maximum value of {T_(r)} satisfying these conditions and T₂ is the minimum value of {T_(r)} satisfying these conditions.

Furthermore, the acts of letting s_(r,k) denote the k^(th) value and adjusting s_(r,k), s_(r,k) is adjusted for each bounding grid point o_(j)(i=1,2,3,4) and bounding period T_(i) (i=1,2) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s).

In yet another aspect, the act of determining G_(ij,NZ) further comprises acts of letting G_(ij,NZ) denote the damping- and site-class-adjusted mean exceedance frequency of s_(NZ) given period T_(i) and location o_(j), and for i=1,2 and j=1,2,3,4, interpolating each curve from “Act d” to determine G_(ij,NZ), with G_(ij,NZ) being calculated according to the following: $G_{i,j,{NZ}} = {{\exp\left( {{\ln\quad G_{i,j,k}} + {{\ln\left( {G_{i,j,{k + 1}}/G_{i,j,k}} \right)}\frac{\left( {s_{NZ} - s_{i,j,k}} \right)}{\left( {s_{i,j,{k + 1}} - s_{i,j,k}} \right)}}} \right)}.}$

Additionally, the act of calculating G_(j,NZ) further comprises acts of letting G_(j,NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location o_(j), and for j=1,2,3,4, and calculating G_(j,NZ) according to the following: $G_{j,{NZ}} = {{\exp\left( {{\ln\quad G_{1,j,{NZ}}} + {{\ln\left( {G_{2,j,{NZ}}/G_{1,j,{NZ}}} \right)}\frac{\ln\left( {\hat{T}/T_{1}} \right)}{\ln\left( {T_{2}/T_{1}} \right)}}} \right)}.}$

In another aspect, G_(NZ) is calculated according to the following: b ₁ =G _(2,NZ) −G _(1,NZ) b ₂ =G _(3,NZ) −G _(1,NZ) b ₃ =G _(4,NZ) −G _(3,NZ) −G _(2,NZ) +G _(1,NZ) b₄=G_(1,NZ) G _(NZ) =b ₁ x ² +b ₂ y ² +b ₃ xy+b ₄.

The present invention further comprises an act of calculating a site-classification coefficient F_(c), including acts of:

-   -   letting F_(a,c) and F_(v,c) denote, respectively, the value of         the site coefficients parameters F_(a) and F_(v) as defined by         the American Society of Civil Engineers (ASCE) for site         classification c and reference period T_(i), where the values of         F_(a,c) and F_(v,c) depend on s, and where c_(r) denotes the         reference site classification; and     -   calculating F_(c) according to one of the following:         ${F_{c} = \frac{F_{a,c}}{F_{a,{cr}}}},{{{and}\quad F_{c}} = {\frac{F_{v,c}}{F_{v,{cr}}}.}}$

Additionally, in the act of calculating F_(c), F_(c) is calculated according to the following:

-   -   when T_(i)≦0.3 sec, or for T_(i)=0.5 sec and c ε {A, AB, B, BC},         ${F_{c} = \frac{F_{a,c}}{F_{a,{cr}}}},$     -   and when T_(i)=0.5 sec and c ε {C, CD, D, DE}, or for T_(i)≧1.0         sec, ${F_{c} = \frac{F_{v,c}}{F_{c,{vr}}}},$         where A, AB, B, BC, C, CD, D, DE represent site soil categories,         relevant to amplification and de-amplification of ground shaking         at a site, relative to how strongly the site would shake in the         same earthquake if it had a different site soil category.

In yet another aspect, in the act of calculating site-classification coefficient F_(c), linear interpolation is used to tabulate values of F_(a) and of F_(v) at s=0.00, 0.01, 0.02, . . . 1.25 g, where g denotes a unit of gravity.

In yet another aspect, the present invention further comprises an act of calculating damping coefficient F_(ζ), including acts of:

-   -   letting B_(S,ζ) and B_(1,ζ) denote, respectively, the value of         the damping coefficients B_(S) and B₁ as defined by ASCE for         viscous damping ratio ζ;     -   letting B_(S,ζr) and B_(1,ζr) denote, respectively, the value of         B_(S,ζ) and B_(1,ζ) for the reference damping ratio ζ_(r); and     -   calculating F_(ζ) according to one of the following:         ${F_{\zeta} = \frac{B_{S,{\zeta\quad r}}}{B_{S,\zeta}}},{{{and}\quad F_{\zeta}} = {\frac{B_{1,{\zeta\quad r}}}{B_{1,\zeta}}.}}$

Additionally, in the act of calculating damping coefficient F_(ζ), for ζ_(r)≦0.05, F_(ζ) is calculated according to the following:

-   -   when T_(i)≦0.3 sec or for T_(i)=0.5 sec and c ε {A, AB, B, BC},         ${F_{\zeta} = \frac{B_{S,{\zeta\quad r}}}{B_{S,\zeta}}},$     -   and when T_(i)=0.5 sec and c ε {C, CD, D, DE}, or for T_(i)≧1.0         sec, $F_{\zeta} = {\frac{B_{1,{\zeta\quad r}}}{B_{1,\zeta}}.}$

The present invention further comprises an act of calculating an estimated annualized lost (EAL) according to the following, EAL=H×PFL,

-   -   wherein H represents the site economic hazard parameter;     -   wherein PFL represents a probable frequent loss, mean loss         conditioned on an occurrence of s_(EBE); and     -   wherein s_(EBE) represents a seismic intensity associated with         an economic-basis-earthquake (EBE), whereby through calculating         the EAL, the EAL can be used to manage seismic risk.

The present invention further comprises an act of estimating the PFL, the PFL representing a point on a mean seismic vulnerability function for a frequent shaking intensity.

Additionally, in the act of estimating PFL, PFL is estimated from a method selected from a group consisting of employing assembly-based vulnerability at intensity level s_(EBE), employing linear assembly-based vulnerability at intensity level s_(EBE), and employing expert opinion.

In yet another aspect, employing assembly-based vulnerability at intensity level s_(EBE) further comprises acts of:

-   -   selecting a ground-motion time history and creating a stochastic         structural model;     -   performing a non-linear time history structural analysis using         the ground-motion time history and the structural model to         determine structural response;     -   assessing probabilistic damage via component fragility functions         evaluated using results from the structural response;     -   assessing loss via probabilistic construction cost-estimation         and results of the assessed probabilistic damage; and     -   repeating steps of selecting the ground-motion time history,         creating the stochastic structural model, performing the         non-linear structural analysis, assessing the probabilistic         damage, assessing the loss to estimate a probability         distribution of loss at S_(EBE), and calculating the loss         statistic PFL.

In yet another aspect, employing linear assembly-based vulnerability at intensity level s_(EBE) further comprises acts of:

-   -   defining a facility using a facility defining parameter, the         facility defining parameter being selected from a group         consisting of location, including latitude and longitude, site         soils, substructure, and structural and, nonstructural         components;     -   using the defined facility to create an inventory of damageable         assemblies and identifying each assembly's engineering demand         parameter (EDP);     -   determining the s_(EBE);     -   analyzing a structure by calculating a structural response to         which each damageable assembly is subjected; and     -   analyzing damage and loss estimates to calculate an expected         total cost to repair the structure according to the following,         ${{PFL} = {\left( {1 + {\overset{\_}{C}}_{OP}} \right){\sum\limits_{k = 1}^{N}{{\overset{\_}{y}}_{k}\left( x_{k} \right)}}}},$     -   where subscript k refers to particular assemblies, N refers to a         number of damageable assemblies, and {overscore (C)}_(or) refers         to contractor's mean overhead-and-profit factor.

In yet another aspect, in the act of calculating a risk analysis parameter, the risk analysis parameter is s_(EBE), where s_(EBE) is the seismic intensity associated with an economic-basis earthquake, and wherein the act of calculating s_(EBE) further comprises acts of:

-   -   a. calculating G_(EBE), where G_(EBE) is an average exceedance         frequency of s_(EBE), and where s_(EBE) is the seismic intensity         associated with an economic-basis earthquake;     -   b. identifying grid points closest to ô for location         ô=({circumflex over (φ)}, {circumflex over (λ)}), where         {circumflex over (φ)} denotes a latitude designation and         {circumflex over (λ)} denotes a longitude designation, the grid         points closest to ô being referred to as the bounding grid         points o_(j);     -   c. considering the period of interest {circumflex over (T)},         determining a period of at least one hazard curve, referred to         as a bounding period T_(i);     -   d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s))         of s_(r) in the hazard curve         G(s_(r)|o_(j),c=c_(r),T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for         each bounding grid point o_(j) and bounding period T_(i) to         account for site classification ĉ and for damping ratio         {circumflex over (ζ)}, where s is seismic intensity and where         adjusting s_(r,k) is done by multiplying s_(r,k) by a         site-classification coefficient F_(c) and a damping coefficient         F_(ζ), where by denoting by s_(k) the k^(th) value of s at the         same location and period after adjusting for site classification         and damping ratio,         G(s _(k) |o _(j) ,c=ĉ,T=T _(i) ,ζ={circumflex over (ζ)})= G(F         _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i)Aζ=ζ): k=1,2, .         . . n _(s);     -   e. calculating s_(ij,EBE), where s_(ij,EBE) denotes the damping-         and site-class-adjusted value of s_(EBE) for period T_(i) and         location o_(j);     -   calculating s_(j,EBE), where s_(j,EBE) denotes the damping- and         site-class-adjusted value of s_(EBE) at the period of interest         {circumflex over (T)} and location o_(j);     -   f. calculating s_(EBE), where s_(EBE) denotes the damping- and         site-class-adjusted value of s_(EBE) at the period of interest         {circumflex over (T)} and location of interest ô, whereby         through calculating s_(EBE), a user may calculate an expected         annualized loss (EAL) according to the following:         EAL=H×PFL,     -   wherein PFL represents a probable frequent loss, mean loss         conditioned on an occurrence of s_(EBE).

Additionally, the act of calculating s_(ij,EBE) further comprises acts of letting s_(ij,k) denote the k^(th) value of s of the damping- and site-class-adjusted hazard curve in “Act d” at period T_(i) and location o_(j), and denoting G_(ij,k)=G(s_(k)|o_(j),c=c_(j),T=T_(i),ζ={circumflex over (ζ)}) from “Act d”, and denoting by s_(ij,EBE) the damping- and site-class-adjusted value of s_(EBE) for period T_(i) and location o_(j), and for i=1,2 and j=1,2,3,4, interpolating each curve from “Act d” to determine s_(ij,EBE), where s_(ij,EBE) is calculated according to the following: $s_{i,j,{EBE}} = {s_{i,j,k} + {\left( {s_{i,j,{k + 1}} - s_{i,j,k}} \right){\frac{\ln\left( {G_{EBE}/G_{i,j,k}} \right)}{\ln\left( {G_{i,j,{k + 1}}/G_{i,j,k}} \right)}.}}}$  k:G _(i,j,k+1) <G _(EBE) ≦G _(i,j,k)

Furthermore, the act of calculating s_(j,EBE) further comprises acts of letting s_(j,EBE) denote the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location ô, and for j=1,2,3,4, calculating s_(j,EBE) according to the following: $s_{j,{EBE}} = {s_{1,j,{EBE}} + {{\ln\left( {\hat{T}/T_{1}} \right)}{\frac{\left( {s_{2,j,{EBE}} - s_{1,j,{EBE}}} \right)}{\ln\left( {T_{2}/T_{1}} \right)}.}}}$

Finally, the act of calculating s_(EBE) further comprises acts of letting s_(EBE) denote the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location of interest ô, and calculating s_(EBE) according to the following: a ₁ =s _(2,EBE) −s _(1,EBE) a ₂ =s _(3,EBE) −s _(1,EBE) a ₃ =s _(4,EBE) −s _(3,EBE) −s _(2,EBE) +s _(1,EBE) a₄=s_(1,EBE) s _(EBE) =a ₁ x ² +a ₂ y ² +a ₃ xy+a ₄.

As can be appreciated by one in the art, the present invention is not limited to a method, but can also be incorporated into other aspects such as a data processing system and a computer program product. Other aspects can be configured to be used on a personal computer, to be Internet-accessible, and to perform all of the acts and operations of the method described herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The objects, features and advantages of the present invention will be apparent from the following detailed descriptions of the preferred aspect of the invention in conjunction with reference to the following drawings where:

FIG. 1 is a table illustrating site-classification coefficient F_(a);

FIG. 2 is a table illustrating site-classification coefficient F_(v);

FIG. 3 is a table illustrating B_(S) and B₁ as defined by the American Society of Civil Engineers (ASCE, 2000);

FIG. 4 is an illustration depicting an Internet-based system for risk management according to the present invention;

FIG. 5 is a block diagram of a general computer system for use with the present invention; and

FIG. 6 is an illustrative diagram of a computer-readable medium aspect of the present invention.

DETAILED DESCRIPTION

The present invention relates to risk management. More specifically, the present invention relates to a method, computer program product, and system for employing readily available, hazard data to estimate an expected repair cost for use in risk management, such as in seismic risk management.

The following description, taken in conjunction with the referenced drawings, is presented to enable one of ordinary skill in the art to make and use the invention and to incorporate it in the context of particular applications. Various modifications, as well as a variety of uses in different applications, will be readily apparent to those skilled in the art, and the general principles defined herein, may be applied to a wide range of aspects. Thus, the present invention is not intended to be limited to the aspects presented, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein. For example, the use of specific numerical values can be altered in some circumstances to achieve the desired result. Furthermore, it should be noted that unless explicitly stated otherwise, the FIGs. included herein are illustrated diagrammatically and without any specific scale, as they are provided as qualitative illustrations of the concept of the present invention.

In order to provide a working frame of reference, first a list of abbreviations is provided as a quick reference provided for the reader. Second, a glossary of terms used in the description and claims is provided. Next, a discussion of various principal aspects of the present invention is provided. Fourth, an introduction is presented to provide the reader with a general understanding of the present invention. Fifth, a discussion is provided to give an understanding of the specific details of the present invention. Finally, a conclusion is provided to summarize key aspects of the present invention.

(1) Abbreviations

-   -   A site classification defined by the American Society of Civil         Engineers (ASCE) (2000);     -   AB site class averaged from adjacent site classes as defined by         ASCE (2000) (i.e., classes A and B);     -   ABV assembly-based vulnerability;     -   B site classification defined by ASCE (2000);     -   BC site class averaged from adjacent site classes as defined by         ASCE (2000) (i.e., classes B and C);     -   B_(S) damping coefficient as defined by ASCE (2000), related to         damped elastic spectral acceleration response in the         acceleration-controlled portion of the response spectrum;     -   B₁ damping coefficient as defined by ASCE (2000), related to         damped elastic spectral acceleration response in the         velocity-controlled portion of the response spectrum;     -   C site classification defined by ASCE (2000);     -   CD site class averaged from adjacent site classes as defined by         ASCE (2000) (i.e., classes C and D);     -   c_(d) mean cost to repair one unit of an assembly from damage         state d;     -   C_(OP) factor applied to total direct construction cost to         account for contractor overhead and profit;     -   c_(r) reference site classification used in the gridded hazard         database;     -   ĉ site classification defined according to ASCE (2000), and         reflects geological characteristics (including the average         shearwave velocity) of the top 30 meters of soil at the facility         in question;     -   D site classification defined by ASCE (2000);     -   d particular value of damage state;     -   D uncertain damage state;     -   DE site class averaged from adjacent site classes as defined by         ASCE (2000) (i.e., classes D and E);     -   DBE design-basis earthquake;     -   EAL expected annualized loss, units in money/time (e.g.         dollars/year);     -   E site classification defined by ASCE (2000);     -   EBE economic-basis earthquake;     -   EDP engineering demand parameter;     -   FEMA Federal Emergency Management Agency;     -   F_(x)(x) cumulative distribution function of a generic uncertain         variable X evaluated at a particular value of that variable,         denoted by x;     -   F_(a) a site coefficient defined by ASCE (2000), related to the         design short-period spectral response acceleration parameter;     -   F_(a,c) value of the site coefficients parameter F_(a) for site         classification c and reference period T_(i), where the value of         F_(a,c) depends on s;     -   F_(a,cr) value of the site coefficients parameter F_(a) for site         classification c_(r) and reference period T_(i);     -   F_(c) site-classification coefficient;     -   F_(v) value of the site coefficients parameter F_(v) from ASCE         (2000), related to the design spectral response acceleration         parameter at one-second;     -   F_(v,c) value of the site coefficients parameter F_(v) from         ASCE (2000) for site classification c and reference period         T_(i);     -   F_(v,cr) value of the site coefficients parameter F_(v)         ASCE (2000) for site classification c_(r) and reference period         T_(i);     -   F_(ζ) damping coefficient;     -   G(s) mean annual frequency of exceeding s, units in 1/time;     -   G(s|Y) mean frequency with which shaking intensity s is         exceeded, given condition(s) denoted by Y;     -   G_(EBE) mean annual exceedance frequency of S_(EBE), units in         1/time;     -   G_(NZ) mean annual exceedance frequency of S_(NZ), units in         1/time;     -   H hazard coefficient, units in 1/time;     -   h story height;     -   L modal excitation;     -   LABV linear assembly-based vulnerability;     -   m level or story in a facility;     -   N_(D) number of possible damage states;     -   n_(s) number of discrete points from the hazard curve shown in         the gridded hazard data for any particular grid point and         period;     -   ô location of a facility in question, denoted by longitude and         latitude;     -   p_(EBE) exceedance probability;     -   PFL probable frequent loss, units in money;     -   PML probable maximum loss, units in money;     -   r generally indicates a reference value;     -   s seismic intensity, units vary depending on selection of s. As         an illustration, units for the damped elastic spectral         acceleration are in distance/time² (e.g., ft/sec²);     -   S_(a) spectral acceleration;     -   s_(r) vector of intensity values in the hazard curve for a         particular grid point and period in the gridded hazard data;     -   s_(r,k) k^(th) value (k=1, 2, . . . n_(s)) of s_(r);     -   S_(EBE) seismic intensity associated with the economic-basis         earthquake;     -   S_(NZ) seismic intensity associated with initiation of loss;     -   T period of vibration;     -   {circumflex over (T)} estimated small-amplitude fundamental         period of vibration of a facility in question;     -   T_(r) reference period of vibration appearing in the gridded         hazard data;     -   T₁ largest value of T_(r) that is less than or equal to         {circumflex over (T)};     -   T₂ smallest value of T_(r) that is greater than {circumflex over         (T)};     -   t_(EBE) planning period;     -   x angular distance in longitude between the site in question and         the nearest grid point whose longitude is less than or equal to         that of the site in question, divided by the longitudinal         spacing between grid points;     -   {circumflex over (x)} the median value of a generic uncertain         variable X;     -   X_(d) threshold level of EDP causing a assembly to reach or         exceed damage state d;     -   y angular distance in latitude between the site in question and         the nearest grid point whose latitude is less than or equal to         that of the site in question, divided by the latitude spacing         between grid points;     -   {overscore (y)}(x) mean cost to repair a damaged assembly;     -   ψ₁ fundamental mode-shape vector;     -   ω₁ fundamental frequency of vibration;     -   ζ. damping ratio;     -   φ_(r) a longitude designation in a map grid point;     -   λ_(r) a latitude designation in a map grid point;     -   {circumflex over (ζ)} damping ratio, a mathematical         characteristic of the idealized structural system of the         facility in question;     -   Φ( ) cumulative standard normal distribution evaluated at the         term in parentheses; and     -   β the logarithmic standard deviation of a generic uncertain         variable X, i.e., the standard deviation of the natural         logarithm of X.

(2) Glossary

The purpose of this glossary is to provide a working frame of reference regarding some of the terms used in the description and claims. The glossary is intended to provide the reader with a general understanding of various terms as they are used in this disclosure, and is not intended to limit the scope of these terms. Rather, the scope of the terms is intended to be construed with reference to this disclosure as a whole and with respect to the claims below. The definitions provided should not be considered limiting to the extent that the terms are known in the art.

Assembly—The term “assembly” refers to a collection of one or more basic building components, assembled in place, and defined according to a standard taxonomic system that is based here on the assembly-numbering system of RS Means, located at 63 Smiths Lane, P.O. Box 800, Kingston, Mass., 02364, United States, extended to account for details of seismic resistance.

Economic-Basis Earthquake—The term “economic-basis earthquake” refers to an event that produces a site shaking with a stated exceedance probability during a stated number of years. As a non-limiting example, the stated exceedance probability is approximately 10% and the stated number of years is approximately 5 years.

Expected Annualized Loss—The term “expected annualized loss” (EAL) refers to an average yearly amount of loss when accounting for the frequency and severity of various levels of loss.

Facility—The term “facility” refers to a building, bridge, or other system comprising a combination of civil, structural, architectural, mechanical, electrical, and/or plumbing components, and can also include furnishings, fixtures, and equipment.

Means—The term “means” as used with respect to this invention generally indicates a set of operations to be performed on a computer, and may represent pieces of a whole program or individual, separable, software modules. Non-limiting examples of “means” include computer program code (source or object code) and “hard-coded” electronics (i.e. computer operations coded into a computer chip). The “means” may be stored in the memory of a computer or on a computer readable medium such as a floppy disk, a CD-ROM, and a flash drive.

Probable Frequent Loss—The term “probable frequent loss” (PFL) refers to the expected value of loss conditioned on the occurrence of an economic-basis earthquake.

Probable Maximum Loss—The term “probable maximum loss” (PML) commonly refers to the mean loss or the 90^(th) percentile of loss given the occurrence of what building codes until recently called the design basis earthquake.

Structure—The term “structure” refers to (1) a system of components in a facility that provide the facility's resistance to loads and deformations, e.g., the facility's beams, columns, floor diaphragms, shearwalls, braces, and connections; or (2) synonym for facility.

(3) Principal Aspects

The present invention has three “principal” aspects. The first is a method for seismic risk management where the method typically operates on a computer system. The second principal aspect is a system for seismic risk management, typically in the form of software and/or manual operations, operated using a data processing system (computer). The third principal aspect is a computer program product. The computer program product generally represents computer-readable code (either source or object code) stored on a computer-readable medium such as an optical storage device, e.g., a compact disc (CD) or digital versatile disc (DVD), or a magnetic storage device such as a floppy disk or magnetic tape. Other, non-limiting examples of computer readable media include hard disks, read only memory (ROM), and flash-type memories. These aspects will be described in more detail below.

(4) Introduction

The present invention includes a method, computer program product, and system for risk management using readily available, gridded hazard data. Although for illustrative purposes the invention described herein largely uses language discussing seismic events and seismic risks, the invention is not limited thereto. The method of the present invention can be applied to risk management for structures exposed to any extraordinary event, non-limiting examples of which include an event such as an earthquake, floods, wind, and blast pressure.

A simplified second-generation performance-based earthquake engineering (PBEE) methodology can be used to estimate two important measures of seismic risk for use in standard financial analyses of real-estate investment opportunities. One is a scenario loss measure called the probable frequent loss (PFL), which is a practical measure of seismic risk that an investor can use in the sensitivity-study portion of his or her financial analysis of a property. The second is expected annualized loss (EAL), which can be reflected in the financial analysis as an operating expense (reflecting amortized seismic risk), and which is approximately proportional to PFL through a constant called the economic hazard coefficient, H. An algorithm is presented here for calculating H and the scenario shaking intensity on which PFL is conditioned, using readily available, U.S. Geological Survey (USGS)-produced hazard data and standard adjustments to account for building location, site soil, damping ratio, and fundamental period. The algorithm is implemented in a database application (and/or software) and the resulting data can be accessed via the Internet (and/or on a user's own computer). The ability to look up H on the Internet (or purchase a software product configured to provide H) and provide a user (e.g., a client) with PFL and EAL could represent a valuable new service to be offered by engineering consultants.

In patent application number 10/862,185 (which is incorporated herein by reference as though fully set forth herein), the Applicant disclosed the methodology for calculating PFL and EAL, given s_(NZ), G_(NZ), s_(EBE), and G_(EBE). The present invention expands upon the prior art by disclosing a methodology to calculate G_(NZ), S_(EBE), G_(EBE) and H.

(5) Description of Specific Embodiments

The present application discloses a method for employing readily available, gridded hazard data to calculate G_(NZ), s_(EBE), G_(EBE) and H for any geographic location and fundamental period (T) within the boundaries of the gridded data. G_(NZ), s_(EBE), G_(EBE), and H are utilized as follows: EAL≈H·PFL,   (3) where $\begin{matrix} {{H \equiv \frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}},} & (4) \end{matrix}$ in which EAL is defined as the expected annualized value of repair cost to a particular facility in a particular location and H is referred to as the site economic hazard coefficient. The PFL is a scenario loss estimate: the estimated mean facility repair cost conditioned on the occurrence of shaking in the economic-basis event, EBE. The EBE is defined as the event producing site shaking with 10% probability of exceedance in 5 years (Compare with the design-basis earthquake, DBE, whose shaking intensity has 10% exceedance probability in 50 yrs.). The site shaking intensity of the EBE is denoted by s_(EBE). G_(EBE) is the mean exceedance frequency at that site of the shaking intensity s_(EBE). G_(NZ) refers to the mean exceedance frequency of the lowest shaking intensity s_(NZ) that would produce a nonnegligible repair cost.

The methodology disclosed herein accounts for period, damping ratio, and site soils using standard American Society of Civil Engineers (ASCE (2000)) adjustments, and geographically interpolates between adjacent grid points to provide the output parameters at the location of interest. ASCE world headquarters are located at 1801 Alexander Bell Drive, Reston, Va. 20191-4400, U.S.A. Although values are taken from ASCE (2000) and used as a reference, it should be clear that other equivalent values from other time periods and other organizations can be used. As can be appreciated by one skilled in the art, the ASCE (2000) values are used for convenience and clarity and are not intended to be limited thereto.

“Gridded hazard data” is sets of mean exceedance frequencies G(s_(r)|φ_(r), λ_(r), c_(r), T_(r), ζ_(r)), where G(s|Y) denotes the mean frequency with which shaking intensity s is exceeded, given condition(s) denoted by Y. In the present invention, it is assumed that intensity s is measured in terms of a parameter that can depend on fundamental period T and damping ratio ζ such as damped elastic spectral acceleration. The gridded hazard data present G at a set of discrete values of s_(r); the set is denoted here by {s_(r)}. It is assumed that the data are available at locations (φ_(r), λ_(r)) ε {(φ_(r), λ_(r))}, i.e., at a set of reference (longitude, latitude) map grid points with grid spacing Δφ and Δλ. The data are presented for a reference site classification c_(r). Period is available at a set of reference values T_(r) ε {T_(r)} and at one reference value of damping ratio ζ_(r).

The U.S. Geological Survey (USGS) has produced the relevant data. The USGS presents mean seismic hazard curves for the U.S., with Δφ=Δλ=0.05° for California and Nevada, and Δφ=Δλ=0.10° for the rest of the U.S.; for c_(r) at the boundary between National Earthquake Hazards Reduction Program (NEHRP) site classifications B and C; and for {T_(r)}={0.0, 0.1, 0.2, 0.3, 0.5, 1.0, 2.0 sec}. In that database, {s_(r)} contains n_(s) values of s, where n_(s) is either 18, 19, or 20, depending on T_(r); and ζ_(r)=5% of critical viscous damping. Although the present methodology is illustrated using these specifics and for locations within California, it does not depend on them, only on the general meaning of gridded hazard data as defined above.

(5.1) Methodology for Calculating G_(NZ), s_(EBE), G_(EBE) and H

Act 1: Query the user for at least one of the following input parameters: location ô=({circumflex over (φ)},{circumflex over (λ)}), site classification ĉ, period of interest {circumflex over (T)}, damping ratio {circumflex over (ζ)}, loss-threshold shaking s_(NZ), planning period t_(EBE), and exceedance probability p_(EBE). Although some users may possess information pertaining to all of the input parameters, default values can be offered. (i.e., {circumflex over (ζ)}=0.05, t_(EBE)=5 yr and p_(EBE)=0.10).

Act 2: Calculate G_(EBE) based on t_(EBE) and p_(EBE) assuming Poisson earthquake arrivals: G _(EBE) =−ln(1−p _(EBE))/t_(EBE)   (5)

Please note, the calculation of G_(EBE) is a function of user selected t_(EBE) and p_(EBE), which assumes Poisson arrivals. If G_(EBE) is readily available, a user could select G_(EBE) directly, making Poison occurrences irrelevant.

Act 3: For location ô=({circumflex over (φ)}, {circumflex over (λ)}), find the four grid points closest to ô. The four grid points closest to ô are referred to as the bounding grid points o₁=(φ₁,λ₁), o₂=(φ₁+Δφ, λ₁), o₃=(φ₁, λ₁+Δλ) and o₄=(φ₁+Δφ,λ₁Δλ)}. o₁, o₂, o₃, o₄ ε {(φ_(r), λ_(r))} such that φ_(min)≦φ₁<{circumflex over (φ)}≦φ₁+Δφ≦φ_(max) and λ_(min)≦λ₁<{circumflex over (λ)}≦λ₁+Δλ≦λ_(max). Calculate x=({circumflex over (φ)}−φ₁)/(Δφ) and y=({circumflex over (λ)}−λ₁)/(Δλ).

Act 4: Considering the period of interest {circumflex over (T)}, determine the periods of the two hazard curves, referred to as the bounding periods, T₁, T₂ ε {T_(r)} such that T₁ is the largest value in {T_(r)} meeting the condition T₁<{circumflex over (T)}, and T₂ is the smallest value in {T_(r)} meeting the condition {circumflex over (T)}≦T₂. If {circumflex over (T)} is a member of {T_(r)} and equal to its minimum value, then T_(i)={circumflex over (T)}, and T₂ is the next larger value in {T_(r)}.

Act 5: Let s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j),c=c_(r),T=T_(i),ζ=ζ_(r)) and adjust s_(r,k) for each bounding grid point o_(j)(j=1,2,3,4) and bounding period T_(i)(i=1,2) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}. Adjusting s_(r,k) is done by multiplying s_(r,k) by the site-classification coefficient F_(c) and the damping coefficient F_(ζ). Both coefficients are detailed below. Thus, denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c={circumflex over (c+EE, T=T ¹ , ζ={circumflex over (ζ)})= G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s),   (6)

Act 6: Let s_(ij,k) denote the k^(th) value of s of the damping- and site-class-adjusted hazard curve produced in Act 5 at period T_(i) and location o_(j). Denote G_(ij,k)=G(s_(k)|o_(j),c=c_(j),T=T_(i),ζ={circumflex over (ζ)}) from Act5. Denote by s_(ij,EBE) the damping- and site-class-adjusted value of s_(EBE) for period T_(i) and location o_(j). For i=1,2 and j=1,2,3,4, interpolate each curve from Act 5 to determine s_(ij,EBE): $\begin{matrix} {{s_{i,j,{EBE}} = {s_{i,j,k} + {\left( {s_{i,j,{k + 1}} - s_{i,j,k}} \right)\frac{\ln\left( {G_{EBE}/G_{i,j,k}} \right)}{\ln\left( {G_{i,j,{k + 1}}/G_{i,j,k}} \right)}}}}{{k\text{:}\quad G_{i,j,{k + 1}}} < G_{EBE} \leq G_{i,j,k}}} & (7) \end{matrix}$

Act 7: Let s_(j,EBE) denote the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location o_(j). For j=1,2,3,4, calculate s_(j,EBE) by: $\begin{matrix} {s_{j,{EBE}} = {s_{1,j,{EBE}} + {{\ln\left( {\hat{T}/T_{1}} \right)}\frac{\left( {s_{2,j,{EBE}} - s_{1,j,{EBE}}} \right)}{\ln\left( {T_{2}/T_{1}} \right)}}}} & (8) \end{matrix}$

Act 8: Let s_(EBE) denote the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location of interest ô. Calculate: a ₁ =s _(2,EBE) −s _(1,EBE) a ₂ =s _(3,EBE) −s _(1,EBE) a ₃ =s _(4,EBE) −s _(3,EBE) −s _(2,EBE) +s _(1,EBE) a ₄=s_(1,EBE) s _(EBE) =a ₁ x ² +a ₂ y ² +a ₃ xy+a ₄   (9)

Act 9: Let G_(ij,NZ) denote the damping- and site-class-adjusted mean exceedance frequency of s_(NZ) given period T_(i) and location o_(j). For i=1,2 and j=1,2,3,4, interpolate each curve from Act 5 to determine G_(ij,NZ). $\begin{matrix} {{G_{i,j,{NZ}} = {\exp\left( {{\ln\quad G_{i,j,k}} + {{\ln\left( {G_{i,j,{k + 1}}/G_{i,j,k}} \right)}\frac{\left( {s_{NZ} - s_{i,j,k}} \right)}{\left( {s_{i,j,{k + 1}} - s_{i,j,k}} \right)}}} \right)}}{{k\text{:}\quad s_{i,j,k}} < s_{NZ} \leq s_{i,j,{k + 1}}}} & (10) \end{matrix}$

Act 10: Let G_(j,NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location o_(j). For j=1,2,3,4, calculate G_(j,NZ) by: $\begin{matrix} {G_{j,{NZ}} = {\exp\left( {{\ln\quad G_{1,j,{NZ}}} + {{\ln\left( {G_{2,j,{NZ}}/G_{1,j,{NZ}}} \right)}\frac{\ln\left( {\hat{T}/T_{1}} \right)}{\ln\quad\left( {T_{2}/T_{1}} \right)}}} \right)}} & (11) \end{matrix}$

Act 11: Let G_(NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location of interest ô. Calculate: b ₁ =G _(2,NZ) −G _(1,NZ) b ₂ =G _(3,NZ) −G _(1,NZ) b ₃ =G _(4,NZ) −G _(3,NZ) −G _(2,NZ) +G _(1,NZ) b₄=G_(1,NZ) G _(NZ) =b ₁ x ² +b ₂ y ² +b ₃ xy+b ₄.   (12)

Act 12: Calculate hazard coefficient H according to the following: $H \equiv {\frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}.}$

Act 13: Return s_(EBE) and H to the user, where through H, the user may then easily calculate EAL.

(5.2) Site Soil Classification Coefficient F_(c)

Act 5 in the foregoing methodology discusses an adjustment to account for site soil classification F_(c). In making the adjustment, let F_(a,c) and F_(v,c) denote, respectively, the value of the site coefficients parameters F_(a) and F_(v) from ASCE (2000) for site classification c and reference period T_(i). The values of F_(a,c) and F_(v,c) depend on s. The ASCE (2000) values of F_(a) and F_(v) are duplicated for illustrative purposes in Table 1 and Table 2, using linear interpolation between values of s. The values were provided by ASCE, 2000, FEMA-356: Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Washington, DC, 490 pp. In the tables, values for site classifications A, B, C, D, and E are as shown in ASCE (2000); values for site classes AB, BC, CD, and DE are averaged from adjacent site classes.

Recall that cr denotes the reference site classification. Note that spectral acceleration is generally in the acceleration-controlled region of the response spectrum for period T≦0.3 sec, and in the velocity-controlled region for periods T≧1.0 sec, and will tend to be in the former region for firm sites and the latter for soft sites at T=0.5 sec. Then: $\begin{matrix} {{F_{c} = \frac{F_{a,c}}{F_{a,{cr}}}},} & (13) \end{matrix}$ for T_(i)≦0.3 sec or for T_(i)=0.5 sec and c ε {A, AB, B, BC}. For T_(i)=0.5 sec and c ε {C, CD, D, DE}, or for T_(i)≧1.0 sec, $\begin{matrix} {F_{c} = {\frac{F_{v,c}}{F_{v,{cr}}}.}} & (14) \end{matrix}$

For convenience, when implementing F_(a) and F_(v) in a database, one can use linear interpolation to tabulate values of F_(a) and of F_(v) at s=0.00, 0.01, 0.02, . . . 1.25 g. When required for a particular value of s and c, one rounds s to the nearest tabulated value and uses the associated value of F_(a) or F_(v).

(5.3) Damping Coefficient F_(ζ)

Act 5 in the foregoing methodology mentions an adjustment to account for damping ratio. In that adjustment, let B_(S,ζ) and B_(1,ζ) denote, respectively, the value of the damping coefficients B_(S) and B₁ from ASCE (2000) (as illustrated in Table 3) for viscous damping ratio ζ. Let B_(S,ζr) and B_(1,ζr) denote, respectively, the value of B_(Sζ) and B_(1,ζ) for the reference damping ratio ζ_(r). Then for ζ_(r)≦0.05, F_(ζ) is given by: $\begin{matrix} {F_{\zeta} = \frac{B_{S,{\zeta\quad r}}}{B_{S,\zeta}}} & (15) \end{matrix}$ for T_(i)≦0.3 sec or for T_(i)=0.5 sec and c ε {A, AB, B, BC}. For T_(i)=0.5 sec and c ε {C, CD, D, DE}, or for T_(i)≧1.0 sec, $\begin{matrix} {F_{\zeta} = \frac{B_{1,{\zeta\quad r}}}{B_{1,\zeta}}} & (16) \end{matrix}$

(5.4) Methodology for Calculating PFL

Equation 3 still requires an estimate of PFL. Application '185 disclosed in detail various methods for estimating PFL. For reference and further clarification, the methods for estimating PFL will be discussed briefly herein. One method for estimating PFL is to perform an assembly-based vulnerability (ABV) analysis at the intensity level S_(EBE), including selection of ground-motion time histories, non-linear time-history structural analysis, damage analysis, loss analysis, and simulation to account for uncertainties in ground motion, mass, damping, force-deformation behavior, component capacity, unit repair cost, and contractor overhead and profit. Another method entails employing en expert opinion in obtaining PFL.

In another approach referred to as linear ABV, the loss analysis can be further simplified by taking advantage of the fact that at low levels of intensity, around S_(EBE), the structural response of the facility might be adequately modeled using linear spectral analysis, thus avoiding the time-consuming construction of a non-linear structural model. Furthermore, only mean loss at S_(EBE) is required, not the damage and not an estimate of uncertainty, allowing avoidance of aspects of ABV that are intended to quantify damage and uncertainty.

(5.4.1) PFL and Linear ABV

A simplified approach called linear assembly-based vulnerability (LABV) can be used to calculate PFL and EAL. LABV includes the following four operations:

-   -   1. Facility definition. To define the facility an individual         must know its location (latitude and longitude) and design,         including site soils, substructure, structural and nonstructural         components. The individual creates an inventory of the         damageable assemblies and identifies the engineering demand         parameter (EDP) (e.g., story drift ratio, member force, etc.)         that would cause damage to each assembly.     -   2. Hazard analysis. The objective of this stage is to determine         the S_(EBE), that is, the intensity measure associated with 10%         exceedance probability in 5 years. This might be parameterized         via any one of several intensity measures. For present purposes,         the 5%-damped elastic spectral acceleration response at the         building's estimated small-amplitude fundamental period of         vibration can be utilized. It can be calculated via software and         adjusted to account for site classification such as by using         F_(a) or F_(v), as appropriate, from the International Building         Code (2000) provided by the International Code Council, located         at 5203 Leesburg Pike, Suite 600, Falls Church, Va., 22041,         United States.     -   3. Structural analysis. In this simplification, the structural         response to which each damageable assembly is subjected is         calculated considering the first-mode spectral response. The         mode shape of a building at its small-amplitude fundamental         period of vibration, T₁, is denoted by ψ₁. Let the modal         excitation and modal mass for the first mode be denoted by L₁         and M₁, respectively. Each damageable assembly is assumed to be         sensitive to an EDP, characteristic of that assembly type, with         a value denoted by x, and which can be calculated as a function         of ψ₁, L₁ and M₁. For example, considering one frame direction,         the EDP for a segment of wallboard partition on the m^(th) story         would be the interstory drift along that wall line, estimated as         $\begin{matrix}         {{x \approx {\frac{s_{EBE}}{\omega_{1}^{2}}\left( \frac{\psi_{1{({m + 1})}} - \psi_{1m}}{h_{m}} \right)\frac{L_{1}}{M_{1}}}},} & (17)         \end{matrix}$     -   where ω₁=2π/T₁, ψ _(1m) refers to the component of the         fundamental mode shape at floor m, and h_(m) refers to the         height of story m.     -   4. Damage and loss analysis. It is assumed that after an         assembly is subjected to a certain EDP, it will be in an         uncertain damage state D, indexed by d=0, 1, 2, . . . N_(D),         where d=0 indicates the undamaged state. It is assumed that the         damage states can be sorted in increasing order, either because         an assembly in damage state d=i+1 must have passed through         damage state i already, or because the effort to restore an         assembly from damage state d=i+1 necessarily restores it from         damage state d=i. The threshold level of EDP causing an assembly         to reach or exceed damage state d is uncertain, and is denoted         by X_(d), with a cumulative distribution function denoted by         F_(Xd)(x). The expected value of the cost to restore a damaged         assembly from damage state d is denoted by c_(d); and can be         calculated by standard construction-cost estimation principles.         Then, given the response x to which an assembly is subjected,         the mean cost to repair the damageable assembly is         $\begin{matrix}         {{{\overset{\_}{y}(x)} = {\sum\limits_{d = 1}^{N_{D}}{c_{d\quad}{p\left\lbrack {D = {\left. d \middle| {EDP} \right. = x}} \right\rbrack}}}},} & (18)         \end{matrix}$     -   where probability $\begin{matrix}         \begin{matrix}         {{p\left\lbrack {D = {\left. d \middle| {EDP} \right. = x}} \right\rbrack} = {1 - {F_{X_{1}}(x)}}} & {d = 0} \\         {= {{F_{X_{d}}(x)} - {F_{X_{d - 1}}(x)}}} & {{1 \leq d < N_{D}},} \\         {= {F_{X_{ND}}(x)}} & {{d = N_{D}},}         \end{matrix} & (19)         \end{matrix}$     -   and where d=0 refers to the undamaged state.

Equation (18) is normalized by the replacement cost of the assembly, and is referred to as a component damage function. The present invention uses the non-normalized form to avoid considering the uncertain replacement cost of the component. That is, because construction contractors estimate repair effort directly in terms of labor hours and dollar costs, it is less prone to error to deal with c_(d) directly, rather than as a fraction of another cost that must also be estimated. Taking all capacities from Beck, J. L.; Porter, K. A.; Shaikhutdinov, R. V.; Au, S. K.; Moroi, T.; Tsukada, Y.; and Masuda, M., 2002, “Impact of Seismic Risk on Lifetime Property Values, Final Report,” Consortium of Universities for Research in Earthquake Engineering, Richmond, Calif., United States, as lognormally distributed, and using the median and logarithmic standard deviations shown there and in Porter, K. A.; Beck, J. L.; and Shaikhutdinov, R. V.; 2002a, “Sensitivity of building loss estimates to major uncertain variables,” Earthquake Spectra, 18 (4), 719-743, so $\begin{matrix} {{{F_{X}(x)} = {\Phi\left( \frac{\ln\left( {x/\hat{x}} \right)}{\beta} \right)}},} & (20) \end{matrix}$

-   -   where {circumflex over (x)} and β vary by assembly type and         damage state, and where Φ( ) denotes the cumulative standard         normal distribution evaluated at the term in parentheses.         Introducing subscript k to index particular assemblies, the         expected total cost to repair the facility with N damageable         assemblies is given by $\begin{matrix}         {{{PFL} = {\left( {1 + {\overset{\_}{C}}_{OP}} \right){\sum\limits_{k = 1}^{N}{{\overset{\_}{y}}_{k}\left( x_{k} \right)}}}},} & (21)         \end{matrix}$         where {overscore (C)}_(OP) refers to contractor's mean         overhead-and-profit factor (typically 15% to 20%). Then EAL is         calculated using Equation 3.

(5.5) Database Implementation

The foregoing methodology was implemented in a Microsoft Access 2003 database, which contains all of the tables and queries necessary to perform the calculations shown above. The database uses the Frankel and Leyendecker gridded hazard data (i.e., Frankel, A. and E. V. Leyendecker, 2001, Uniform Hazard Response Spectra and Seismic Hazard Curves for the United States, CD-ROM, US Geological Survey, Menlo Park, Calif.) and was tested on the site shaking hazard for the case-study buildings discussed in application '185. Use of the database produced the same results as were produced manually in application '185.

Although the methodology was implemented in a Microsoft Access 2003 database, Access 2003 is not strictly required as any suitable database with table and query functions will work, non-limiting examples of which include Extensible Markup Language (XML), an international standard language for expressing table structure and contents, and Structured Query Language (SQL), an international standard query language. A macro that runs the queries in the desired order has been created in the form of Visual Basic source code. Access to the database could be readily offered via the Internet for low-cost, on-demand use by engineering consultants and others.

(5.6) Data Processing System

The present invention also comprises a system for risk management. Through use of a data processing system, the risk management method can be incorporated into a Website. The Website allows users to access it over the Internet so that users can interact with the system or download a copy of a computer program product containing computer executable instructions for operating the method described herein. Alternatively, the system can be loaded onto a home computer or laptop computer where a user interacts with the computer to manage risk.

FIG. 4 illustrates a data processing system 400 (e.g., server computer) incorporating the operations of the method described above. The method utilizes the data processing system 400 for storing computer executable instructions as components for causing a processor to carry out the operations. The data processing system 400 may be accessible by a user's computer 402 over the Internet 404 through an Internet-accessing user interface (e.g., web page).

In another aspect, the computer executable instructions for causing a processor to carry out the operations of the method can be loaded onto the user's computer 402, personal digital assistant, or any other computing device. In this aspect, the user's computer 402 functions as the data processing system 400.

A block diagram depicting the components of the data processing system 400 used in the present invention is provided in FIG. 5. The data processing system 400 comprises an input 500 for receiving information from a user. Information received may include input from devices such as scanners, keypads, keyboards, microphone, other peripherals such as storage devices, other programs, etc. The input 500 may include multiple “ports.” An output 502 is connected with the processor for providing information for transmission to other data processing systems, to storage devices, to display devices such as monitors, to generating information necessary for delivery, and to other mechanisms for presentation in user-readable forms. Output may also be provided to other devices or other programs, e.g., to other software modules, for use therein. The input 500 and the output 502 are both coupled with a processor 504, which may be a general-purpose computer processor or a specialized processor designed specifically for use with the present invention. The processor 504 is coupled with a memory 506 to permit storage of data and software to be manipulated by commands to the processor.

(5.7) Computer Program Product

An illustrative diagram of a computer program product embodying the present invention is depicted in FIG. 6. As a non-limiting example, the computer program product is depicted as either a floppy disk 600 or an optical disk 602. However, as mentioned previously, the computer program product generally represents computer-readable code stored on any compatible computer-readable medium for causing a processor to carry out the operations of the above described method.

(6) Conclusion

Previous work showed that two useful real-estate investment decision parameters, the probable frequent loss (PFL) and expected annualized loss (EAL) can be readily calculated by engineering consultants and employed by real-estate investors to reflect seismic risk in their financial analysis of an investment opportunity. The present application discloses how one can create a GIS database of the constant of proportionality, H, called the economic hazard coefficient, from readily available gridded hazard data produced by the U.S. Geological Survey. The database also includes the shaking intensity, s_(EBE), needed for determining PFL. The database can be utilized by engineering consultants via the Internet, or on any other computer readable medium.

Finally, the present application discloses how formal decision analysis can be used in the investment decision process to account for real estate market risk, future earthquake losses, and the investor's risk attitude, and to choose among competing risk-management alternatives based on the maximum certainty equivalent. The decision-analysis approach requires the additional information of variance of market value and the decision-maker's risk tolerance. Variance of earthquake repair costs is not needed, since it makes a negligible contribution in the decision analysis compared with the uncertainty in market conditions. 

1. A method for obtaining a risk analysis parameter using a data processing system having an input and an output, the method comprising acts of: a. querying a user to input at least one of the following input parameters into the input: i. location ô=({circumflex over (φ)},{circumflex over (λ)}), where ô denotes a longitude and latitude of a facility in question; ii. site classification ĉ, where the site classification reflects geological characteristics of soil at the facility in question; iii. period of interest {circumflex over (T)}, where {circumflex over (T)} denotes a user's estimate of the small-amplitude fundamental period of vibration of the facility in question; iv. damping ratio {circumflex over (ζ)} where the damping ration is a mathematical characteristic of an idealized structural system of the facility in question; v. loss-threshold shaking s_(NZ), where the loss-threshold shaking is a minimum level of shaking at which nonnegligible damage occurs; vi. planning period t_(EBE), where the planning period is the period an investor uses in a financial analysis for the facility in question; and vii. exceedance probability p_(EBE), where the exceedance probability is the probability that earthquake shaking of intensity s_(EBE) or greater will occur during planning period t_(EBE); b. calculating a risk analysis parameter based upon at least one of the input parameters; c. outputting the risk analysis parameter through the output, whereby through the risk analysis parameter, a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, average loss conditioned on a seismic intensity associated with an economic-basis earthquake, and H represents an economic hazard coefficient.
 2. A method for obtaining a risk analysis parameter as set forth in claim 1, wherein in the act of calculating a risk analysis parameter, the risk analysis parameter is hazard coefficient H, and the act of calculating hazard coefficient H further comprising acts of: a. calculating G_(EBE), where G_(EBE) is an average exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake; b. identifying grid points closest to ô for location ô=({circumflex over (φ)}, {circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the grid points closest to ô being referred to as the bounding grid points o_(j); c. considering the period of interest {circumflex over (T)}, determining a period of at least one hazard curve, referred to as a bounding period T_(i); d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j),c=c_(r),T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for each bounding grid point o_(j) and bounding period T_(i) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s); e. letting G_(ij,NZ) denote the damping- and site-class-adjusted average exceedance frequency of s_(NZ) given period T_(i) and location o_(j), and for each T_(i) and o_(j), interpolating each curve from “Act d” to determine G_(ij,NZ); f. letting G_(j,NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location o_(j), and for each j, calculating G_(j,NZ); g. letting G_(NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location of interest ô, and calculating G_(NZ); h. calculating risk analysis parameter H (i.e., hazard coefficient) according to the following: ${H \equiv \frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}},$ whereby through H, a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, mean loss conditioned on an occurrence of s_(EBE).
 3. A method for obtaining a risk analysis parameter as set forth in claim 2, wherein the act of calculating G_(EBE), G_(EBE) is calculated based on t_(EBE) and p_(EBE) assuming Poisson earthquake arrivals according to the following: G _(EBE) =−ln(1−p _(EBE))/t _(EBE), where G_(EBE) is a mean annual exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake.
 4. A method for obtaining a risk analysis parameter as set forth in claim 2, wherein the act of identifying grid points closest ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), further comprises acts of identifying four grid points closest to ô for location ô=({circumflex over (φ)}, {circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the four grid points closest to ô being referred to as the bounding grid points o₁=(φ₁,λ₁), o₂=(φ₁+Δφ, λ₁), o₃=(φ₁, λ₁+Δλ) and o₄=(φ₁+Δφ,λ₁+Δλ)}, where o₁, o₂, o₃, o₄ ε {(φ_(r), λ_(r))} such that φ_(min)≦φ₁<{circumflex over (φ)}≦φ₁+Δφ≦φ_(max) and λ_(min)≦λ₁<{circumflex over (λ)}≦λ₁+Δλ≦λ_(max), and calculating x and y according to the following: x=({circumflex over (φ)}−φ₁)/(Δφ) and y=({circumflex over (λ)}−λ₁)/(Δλ).
 5. A method for obtaining a risk analysis parameter as set forth in claim 2, wherein the act of determining a period of at least one hazard curve, further comprises an act of determining periods of two hazard curves, referred to as bounding periods, T₁, T₂ ε {T_(r)}, such that T₁<{circumflex over (T)}≦T₂, or T₁={circumflex over (T)} if {circumflex over (T)} is in {T_(r)} and is equal to its minimum value, and T₁ is the maximum value of {T_(r)} satisfying these conditions and T₂ is the minimum value of {T_(r)} satisfying these conditions.
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 20. A method for obtaining a risk analysis parameter as set forth in claim 1, wherein in the act of calculating a risk analysis parameter, the risk analysis parameter is s_(EBE), where s_(EBE) is the seismic intensity associated with an economic-basis earthquake, and wherein the act of calculating s_(EBE) further comprises acts of: a. calculating G_(EBE), where G_(EBE) is an average exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake; b. identifying grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)}denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the grid points closet to ô being referred to as the bounding grid points o_(j); c. considering the period of interest {circumflex over (T)}, determining a period of at least one hazard curve, referred to as a bounding period T_(i); d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j),c=c_(r),T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for each bounding grid point o_(j) and bounding period T_(i) to account for site classification ĉ and for damping ratio {circumflex over (λ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(λ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i), ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s); e. calculating s_(ij,EBE), where s_(ij, EBE) denotes the damping- and site-class-adjusted value of s_(EBE) for period T_(i) and location o_(j); calculating s_(j,EBE), where s_(j,EBE) denotes the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location o_(j); f. calculating s_(EBE), where s_(EBE) denotes the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location of interest ô, whereby through calculating s_(EBE), a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probably frequent loss, mean loss conditioned on an occurrence of s_(EBE).
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 39. A system for obtaining a risk analysis parameter, the system being configured to perform operations of: a. querying a user to input at least one of the following input parameters into the input: i. location ô=({circumflex over (φ)},{circumflex over (λ)}), where ô denotes a longitude and latitude of a facility in question; ii. site classification ĉ, where the site classification reflects geological characteristics of soil at the facility in question; iii. period of interest {circumflex over (T)}, where {circumflex over (T)} denotes a user's estimate of the small-amplitude fundamental period of vibration of the facility in question; iv. damping ration {circumflex over (ζ)}, where the damping ration is a mathematical characteristic of an idealized structural system of the facility in question; v. loss-threshold shaking s_(NZ), where the loss-threshold shaking is a minimum level of shaking at which nonnegligible damage occurs; vi. planning period t_(EBE), where the planning period is the period and investor uses in a financial analysis for the facility in question; and vii. exceedance probability p_(EBE), where the exceedance probability is the probability that earthquake shaking of intensity s_(EBE) or greater will occur during planning period t_(EBE); b. calculating a risk analysis parameter based upon at least one of the input parameters; c. outputting the risk analysis parameter, whereby through the risk analysis parameter, a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, average loss conditioned on a seismic intensity associated with and economic-basis earthquake, and H represents an economic hazard coefficient.
 40. A system for obtaining a risk analysis parameter as set forth in claim 39, wherein in the operation of calculating a risk analysis parameter, the risk analysis parameter is hazard coefficient H, and the operation of calculating hazard coefficient H further comprising operations of: a. calculating G_(EBE), where G_(EBE) is an average exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake; b. identifying grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the grid points closest to ô being referred to as the bounding grid points o_(j); c. considering the period of interest {circumflex over (T)}, determining a period of at least one hazard curve, referred to as a bounding period T_(i); d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j), c=c_(r), T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for each bounding grid point o_(j) and bounding period T_(i) to account for site classification ĉ and for damping ration {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) , c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) , c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s); e. letting G_(ij,NZ) denote the damping- and site-class-adjusted average exceedance frequency of s_(NZ) given period T_(i) and location o_(j), and for each T_(i) and o_(j), interpolating each curve from “Act d” to determine G_(ij,NZ); f. letting G_(j,NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location o_(j), and for each j, calculating G_(j,NZ); g. letting G_(NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location of interest ô, and calculating G_(NZ); h. calculating risk analysis parameter H (i.e., hazard coefficient) according to the following: ${H \equiv \frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}},$ whereby through H, a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, mean loss conditioned on an occurrence of s_(EBE).
 41. A system for obtaining a risk analysis parameter as set forth in claim 40, wherein the operation of calculating G_(EBE), G_(EBE) is calculated based on t_(EBE) and p_(EBE) assuming Poisson earthquake arrivals according to the following: G _(EBE) =−ln(1−p _(EBE))/t _(EBE), where G_(EBE) is a mean annual exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake.
 42. A system for obtaining a risk analysis parameter as set forth in claim 40, wherein the operation of identifying grid points closest ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), further comprises operations of identifying four grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the four grid points closest to ô being referred to as the bounding grid points o₁=(φ₁, λ₁), o₂=(φ₁+Δφ, λ₁), o₃=(φ₁, λ₁+Δλ) and o₄=(φ₁+Δφ,λ₁+Δλ)}, where o₁, o₂, o₃, o₄ ε {(φ_(r), λ_(r))} such that φ_(min)≦φ₁<{circumflex over (φ)}≦φ₁+Δφ≦φ_(max) and λ_(min)≦λ₁<{circumflex over (λ)}≦λ₁+Δλ≦λ_(max), and calculating x and y according to the following: x=({circumflex over (φ)}−φ₁)/(Δφ) and y=({circumflex over (λ)}−λ₁)/(Δλ).
 43. A system for obtaining a risk analysis parameter as set forth in claim 40, wherein the operation of determining a period of at least one hazard curve, further comprises an operation of determining periods of two hazard curves, referred to as bounding periods, T₁, T₂ ε {T_(r)}, such that T₁<{circumflex over (T)}≦T₂, or T₁={circumflex over (T)} if {circumflex over (T)} is in {T_(r)} and is equal to its minimum value, and T₁ is the maximum value of {T_(r)} satisfying these conditions and T₂ is the minimum value of {T_(r)} satisfying these conditions.
 44. A system for obtaining a risk analysis parameter as set forth in claim 40, wherein the operations of letting s_(r,k) denote the k^(th) value and adjusting s_(r,k), s_(r,k) is adjusted for each bounding grid point o_(j)(=1,2,3,4) and bounding period T_(i)(i=1,2) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) , c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s).
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 58. A system for obtaining a risk analysis parameter as set forth in claim 39, wherein in the operation of calculating a risk analysis parameter, the risk analysis parameter is s_(EBE), where s_(EBE) is the seismic intensity associated with an economic-basis earthquake, and wherein the operation of calculating s_(EBE) further comprises operations of: a. calculating G_(EBE), where G_(EBE) is an average exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake; b. identifying grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the grid points closest to ô being referred to as the bounding grid points o_(j); c. considering the period of interest {circumflex over (T)}, determining a period of at least one hazard curve, referred to as a bounding period T_(i); d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j),c=c_(r),T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for each bounding grid point o_(j) and bounding period T_(i) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i), ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i), ζ=ζ_(r)): k=1,2, . . . n _(s); e. calculating s_(ij,EBE), where s_(ij,EBE) denotes the damping- and site-class-adjusted value of s_(EBE) for period T_(i) and location o_(j); calculating s_(j,EBE), where s_(j,EBE) denotes the damping- and site-class-adjusted value of s_(EBE) at the period of interest T and location o_(j); f. calculating s_(EBE), where s_(EBE) denotes the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location of interest ô, whereby through calculating s_(EBE), a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, mean loss conditioned on an occurrence of s_(EBE).
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 77. A computer program product for obtaining a risk analysis parameter, the computer program product comprising computer-readable instruction means stored on a computer readable medium for causing a computer to perform operations of: a. querying a user to input at least one of the following input parameters into the input: i. location ô=({circumflex over (φ)},{circumflex over (λ)}), where ô denotes a longitude and latitude of a facility in question; ii. site classification ĉ, where the site classification reflects geological characteristics of soil at the facility in question; iii. period of interest {circumflex over (T)}, where {circumflex over (T)} denotes a user's estimate of the small-amplitude fundamental period of vibration of the facility in question; iv. damping ratio {circumflex over (ζ)}, where the damping ration is a mathematical characteristic of an idealized structural system of the facility in question; v. loss-threshold shaking s_(NZ), where the loss-threshold shaking is a minimum level of shaking at which nonnegligible damage occurs; vi. planning period t_(EBE), where the planning period is the period an investor uses in a financial analysis for the facility in question; and vii. exceedance probability p_(EBE), where the exceedance probability is the probability that earthquake shaking of intensity s_(EBE) or greater will occur during planning period t_(EBE); b. calculating a risk analysis parameter based upon at least one of the input parameters; c. outputting the risk analysis parameter, whereby through the risk analysis parameter, a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, average loss conditioned on a seismic intensity associated with an economic-basis earthquake, and H represents an economic hazard coefficient.
 78. A computer program product for obtaining a risk analysis parameter as set forth in claim 77, wherein in the operation of calculating a risk analysis parameter, the risk analysis parameter is hazard coefficient H, and the operation of calculating hazard coefficient H further comprising operations of: a. calculating G_(EBE), where G_(EBE) is an average exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake; b. identifying grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the grid points closest to ô being referred to as the bounding grid points o_(j); c. considering the period of interest {circumflex over (T)}, determining a period of at least one hazard curve, referred to as a bounding period T_(i); d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j),C=c_(r),T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for each bounding grid point o_(j) and bounding period T_(i) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s); e. letting G_(ij,NZ) denote the damping- and site-class-adjusted average exceedance frequency of s_(NZ) given period T_(i) and location o_(j), and for each T_(i) and o_(j), interpolating each curve from “Act d” to determine G_(ij,NZ); f. letting G_(j,NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location o_(j), and for each j, calculating G_(j,NZ); g. letting G_(NZ) denote the damping- and site-class-adjusted value of G_(NZ) at the period of interest {circumflex over (T)} and location of interest ô, and calculating G_(NZ); h. calculating risk analysis parameter H (i.e., hazard coefficient) according to the following: ${H \equiv \frac{G_{NZ}}{\ln\left( {G_{NZ}/G_{EBE}} \right)}},$ whereby through H, a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, mean loss conditioned on an occurrence of s_(EBE).
 79. A computer program product for obtaining a risk analysis parameter as set forth in claim 78, wherein the operation of calculating G_(EBE), G_(EBE) is calculated based on t_(EBE) and p_(EBE) assuming Poisson earthquake arrivals according to the following: G _(EBE) =<ln(1−p_(EBE))/t _(EBE), where G_(EBE) is a mean annual exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake.
 80. A computer program product for obtaining a risk analysis parameter as set forth in claim 78, wherein the operation of identifying grid points closest ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), further comprises operations of identifying four grid points closest to ô for location ô=({circumflex over (φ)}, {circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the four grid points closest to ô being referred to as the bounding grid points o₁=(φ₁, λ₁), o₂=(φ₁+Δφ, λ₁), o₃=(φ₁, λ₁+Δλ) and o₄=(φ₁+Δφ, λ₁+Δλ)}, where o₁, o₂, o₃, o₄ ε {(φ_(r), λ_(r))} such that φ_(min)≦φ₁<{circumflex over (φ)}≦φ₁+Δφ≦φ_(max) and λ_(min)≦λ₁<{circumflex over (λ)}≦λ₁+Δλ≦λ_(max), and calculating x and y according to the following: x=({circumflex over (φ)}−φ₁)/(Δφ) and y=({circumflex over (λ)}−λ₁)/(Δλ).
 81. A computer program product for obtaining a risk analysis parameter as set forth in claim 78, wherein the operation of determining a period of at least one hazard curve, further comprises an operation of determining periods of two hazard curves, referred to as bounding periods, T₁, T₂ ε {T_(r)}, such that T₁<{circumflex over (T)}≦T₂, or T₁={circumflex over (T)} if {circumflex over (T)} is in {T_(r)} and is equal to its minimum value, and T₁ is the maximum value of {T_(r)} satisfying these conditions and T₂ is the minimum value of {T_(r)} satisfying these conditions.
 82. A computer program product for obtaining a risk analysis parameter as set forth in claim 78, wherein the operations of letting s_(r,k) denote the k^(th) value and adjusting s_(r,k), s_(r,k) is adjusted for each bounding grid point o_(j) (j=1,2,3,4) and bounding period T_(i)(i=1,2) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s).
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 96. A computer program product for obtaining a risk analysis parameter as set forth in claim 77, wherein in the operation of calculating a risk analysis parameter, the risk analysis parameter is s_(EBE), where s_(EBE) is the seismic intensity associated with an economic-basis earthquake, and wherein the operation of calculating s_(EBE) further comprises operations of: a. calculating G_(EBE), where G_(EBE) is an average exceedance frequency of s_(EBE), and where s_(EBE) is the seismic intensity associated with an economic-basis earthquake; b. identifying grid points closest to ô for location ô=({circumflex over (φ)},{circumflex over (λ)}), where {circumflex over (φ)} denotes a latitude designation and {circumflex over (λ)} denotes a longitude designation, the grid points closest to ô being referred to as the bounding grid points o_(j); c. considering the period of interest {circumflex over (T)}, determining a period of at least one hazard curve, referred to as a bounding period T_(i); d. letting s_(r,k) denote the k^(th) value (k=1, 2, . . . n_(s)) of s_(r) in the hazard curve G(s_(r)|o_(j),c=c_(r),T=T_(i),ζ=ζ_(r)) and adjusting s_(r,k) for each bounding grid point o_(j) and bounding period T_(i) to account for site classification ĉ and for damping ratio {circumflex over (ζ)}, where s is seismic intensity and where adjusting s_(r,k) is done by multiplying s_(r,k) by a site-classification coefficient F_(c) and a damping coefficient F_(ζ), where by denoting by s_(k) the k^(th) value of s at the same location and period after adjusting for site classification and damping ratio, G(s _(k) |o _(j) ,c=ĉ,T=T _(i),ζ={circumflex over (ζ)})=G(F _(c) F _(ζ) s _(r,k) |o _(j) ,c=c _(r) ,T=T _(i),ζ=ζ_(r)): k=1,2, . . . n _(s); e. calculating s_(ij,EBE), where s_(ij,EBE) denotes the damping- and site-class-adjusted value of s_(EBE) for period T_(i) and location o_(j); calculating s_(j,EBE), where s_(j,EBE) denotes the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location o_(j); f. calculating s_(EDE), where s_(EBE) denotes the damping- and site-class-adjusted value of s_(EBE) at the period of interest {circumflex over (T)} and location of interest ô, whereby through calculating s_(EBE), a user may calculate an expected annualized loss (EAL) according to the following: EAL=H×PFL, wherein PFL represents a probable frequent loss, mean loss conditioned on an occurrence of s_(EBE).
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